Tarea 14, Análisis real

Due November 13

Problem 1 (Bessel inequality)

Let $latex X$ be a separable Hilbert space and $latex \{v_k\}$ an orthonormal set (may be finite). Then

$latex \displaystyle \sum_k |(x,v_k)|^2 \le ||x||^2.$

Problem 2

Consider the function $latex ||\cdot||_\infty:l^2\to[0,\infty)$ given by

$latex ||(a_n)||_\infty = \sup\{|a_n|:n\ge 1\}$.

1. $latex ||\cdot||_\infty$ is a norm in $latex l^2$


2. Is $latex ||\cdot||_\infty$ equivalent to the $latex l^2$-norm?


3. Is $latex (l^2,||\cdot||_\infty)$ complete?

Problem 3

Let $latex X$ be a separable Hilbert space and $latex Y$ a closed subspace. Then $latex Y$ is separable.

Problem 4

Let $latex X$ be a separable inner product space and $latex \bar X$ its completion. Then $latex \bar X$ is a separable Hilbert space.

Problem 5

Let $latex \phi\in X'$, where $latex X$ is a Hilbert space. Then $latex \ker\phi$ is a closed subspace of $latex X$ of codimension 1.